On a generalization of Rédei's theorem (Q1882115)

Let \(p\) be a prime and consider a set \(U\) of \(p\) points in AG\((2,p)\). Let \(N\) be the number of distinct ``directions'' or slopes (including \(\infty\)) of all lines resulting from connecting any two points in \(U\). The case \(p=2\) is trivial, and the case \(p=3\) is easy, so we can assume \(p\geq5\). Then the following cases occur (i) \(U\) is a line; (ii) \(N={p+3\over 2}\) and \(U\) is an example given by \textit{L. Lovász} and \textit{A. Schrijver} [Stud. Sci. Math. Hung. 16, 449--454 (1981; Zbl 0535.51009)]; (iii) \(N\geq\lfloor2{p-1\over 3}\rfloor+1\) (the author's main result). A simple example with \(N=2{p-1\over 3}+2\) for \(p\equiv1\bmod3\) is given. This shows that the bound is not far from being sharp. Some remarks are made for the case when \(p\) is a prime power, rather than just a prime. For the proof the statement is translated into an algebraic setting, using an interpolation polynomial over the field with \(p\) elements. Another reformulation uses permutation polynomials.